Finite Group Databasebeta

← #14327713

Subgroups — #14327713

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91781C211yesmaximal, characteristic, complemented
2C2×C5×#892C211yesmaximal, characteristic, complemented
3C2×C4×#1203C211yesmaximal, characteristic
4C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
5#11748908C212yesmaximal
6#11748834C211yesmaximal, characteristic, complemented
7C4×#6668C212yesmaximal, complemented
8#11748799C211yesmaximal, characteristic, complemented
9#11748786C211yesmaximal, characteristic, complemented
10#11748730C212yesmaximal, complemented
11#11748724C212yesmaximal, complemented
12#10613972—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C4×C5×D3C2211yescharacteristic
15C22×#1203C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×#6714C2212yes—
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4×Dic15C2212yes—
20#91781—24no—
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2212yes—
24C5×#892—24no—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2212yes—
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×#6668C2212yes—
29C2×#6668C2212yes—
30C2×#6668C2211yescharacteristic
31C2×#6667C2212yes—
32C2×#6667C2211yescharacteristic
33C2×(C2×D3)⋊Dic5C2212yes—
34C2×(C2×D3)⋊Dic5C2211yescharacteristic
35C4×#1203C418yescomplemented
36C3×#3963C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#73638—51nomaximal, Hall
40C2×C5×#117—31no—
41C2×C4×C22⋊D5—31no—
42C2×#4038—31no—
43#65501—31no—
44#65276—31no—
45#65265—31no—
46C4⋊C4×Dic5D311yescharacteristic, complemented
47#65246—32no—
48#65233—32no—
49#65229—32no—
50C4×#683—32no—
51C23×D15C2311yescharacteristic
52C23×C5×D3C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57C2×C4×D15—22no—
58C2×C4×D15—22no—
59C2×C4×D15C2×C414yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2311yescharacteristic
62C2×C4×C5×D3—22no—
63C2×C4×C5×D3—22no—
64C2×C4×C5×D3C2×C414yes—
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×#1203—22no—
69C2×#1203—22no—
70C2×#1203C2×C414yes—
71C3×C5×C4⋊C4D414yescomplemented
72#6714—24no—
73C4⋊Dic15—22no—
74C4×Dic15—24no—
75C5×(C2×D3)⋊C4—24no—
76C5×C4⋊Dic3—22no—
77C4×C5×Dic3—24no—
78C3×C4⋊Dic5—22no—
79C3×C4×Dic5—24no—
80C3×C4×Dic5C2×C414yes—
81#6668—24no—
82#6668—42no—
83#6668C2×C414yes—
84#6667—42no—
85#6667C2×C414yes—
86(C2×D3)⋊Dic5—42no—
87(C2×D3)⋊Dic5C2×C414yes—
88C2×#892—51no—
89C2×#892D511yescharacteristic, complemented
90C2×C4×D12—51no—
91C3×C4×C4⋊C4—51no—
92#4987—51no—
93#4987—51no—
94#4939—52no—
95#4935—51no—
96C4×C4⋊Dic3—52no—
97#4670—52no—
98#4670—52no—
99C23×C4×C5—31no—
100C22×C22⋊D5—31no—
101C22×C4×D5—31no—
102C5×#117—64no—
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C5×C22⋊C4—32no—
106C2×C42×C5—32no—
107C2×C22⋊Dic5—31no—
108C2×C22⋊Dic5—32no—
109C4×C22⋊D5—38no—
110C2×#685—31no—
111C2×#685—32no—
112C2×#685—32no—
113C2×C4⋊Dic5—31no—
114C2×C4⋊Dic5C2×D311yescharacteristic
115C2×#683—31no—
116C2×#683—32no—
117C2×#683—32no—
118C2×C4×Dic5—32no—
119C2×C4×Dic5C2×D312yes—
120C2×C4×Dic5C2×D311yescharacteristic
121#4038—64no—
122#3963—32no—
123#3963—32no—
124#3963C2×D312yes—
125#2354—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×D15—21no—
128C22×D15—21no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15—21no—
132C22×D15—21no—
133C22×D15C22×C412yes—
134C22×C5×D3—21no—
135C22×C5×D3—21no—
136C22×C5×D3—21no—
137C22×C5×D3—21no—
138C22×C5×D3—21no—
139C22×C5×D3—21no—
140C22×C5×D3C22×C412yes—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5C22×C412yes—
144C2×C3×C4×C5C2×D414yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4112yes—
147C2×Dic15—21no—
148C2×Dic15—21no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15C22×C412yes—
153C4×D15—28no—
154C2×C5×Dic3—21no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3C22×C412yes—
160C4×C5×D3—28no—
161C2×C3×Dic5—21no—
162C2×C3×Dic5—21no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—22no—
166C2×C3×Dic5C22×C412yes—
167C2×C3×Dic5C22×C412yes—
168C2×C3×Dic5#4112yes—
169C2×C3×Dic5#4112yes—
170#1203—28no—
171C22×D12—51no—
172C22×C4×D3—51no—
173C22×C4×D3C2×D511yescharacteristic
174C2×C3×C4⋊C4—51no—
175C2×C3×C4⋊C4C2×D511yescharacteristic
176C2×C3×C42—51no—
177C2×C3×C42—52no—
178C2×(C2×D3)⋊C4—51no—
179C2×(C2×D3)⋊C4—51no—
180C2×(C2×D3)⋊C4—52no—
181C2×(C2×D3)⋊C4—52no—
182C2×(C2×D3)⋊C4—52no—
183C2×(C2×D3)⋊C4C2×D512yes—
184C2×C4⋊Dic3—51no—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3—52no—
187C2×C4⋊Dic3—52no—
188C2×C4⋊Dic3C2×D511yescharacteristic
189C2×C4×Dic3—52no—
190C2×C4×Dic3C2×D512yes—
191#892—24no—
192#892—104no—
193C4×D12—58no—
194C3×#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C24×C5—31no—
198C23×D5—31no—
199C22×C4×C5—31no—
200C22×C4×C5—34no—
201C22×C4×C5—32no—
202C22×C4×C5—62no—
203C22×C4×C5—62no—
204C22×C4×C5C22×D312yes—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—34no—
207C2×C22⋊D5—62no—
208C2×C22⋊D5—62no—
209C22×Dic5—31no—
210C22×Dic5—32no—
211C22×Dic5C22×D312yes—
212C22×Dic5C22×D311yescharacteristic
213C22×Dic5C22×D311yescharacteristic
214C2×C4×D5—34no—
215C2×C4×D5—62no—
216C2×C4×D5—62no—
217C5×C4⋊C4—62no—
218C5×C4⋊C4D1214yescomplemented
219C5×C22⋊C4—64no—
220C42×C5—64no—
221C22⋊Dic5—34no—
222C22⋊Dic5—122no—
223#685—34no—
224#685—64no—
225#685—122no—
226C4⋊Dic5—22no—
227C4⋊Dic5—62no—
228#683—34no—
229#683—64no—
230#683—122no—
231C4×Dic5—24no—
232C4×Dic5—64no—
233C4×Dic5C4×D314yes—
234C2×C4×D4—151no—
235C2×#117—31no—
236C2×#117—151no—
237#390—152no—
238#388—152no—
239#388—152no—
240#384—151no—
241#380—151no—
242#380—151no—
243C4×C4⋊C4—51no—
244C4×C4⋊C4—152no—
245C22×C3×C5C23×C411yescharacteristic
246C22×C3×C5C22×D411yescharacteristic
247C22×C3×C5C2×#4111yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C22×C3×C5C2×#4111yescharacteristic
250C22×C3×C5H(5,𝔽2)11yescharacteristic
251C22×C3×C52−1+411yescharacteristic
252C2×D15—22no—
253C2×D15—22no—
254C2×D15—22no—
255C2×D15—41no—
256C2×D15—41no—
257C2×D15—41no—
258C2×D15—41no—
259C2×C5×D3—22no—
260C2×C5×D3—22no—
261C2×C5×D3—22no—
262C2×C5×D3—41no—
263C2×C5×D3—41no—
264C2×C5×D3—41no—
265C2×C5×D3—41no—
266C3×C4×C5—24no—
267C3×C4×C5C4×D414yes—
268Dic15—22no—
269Dic15—42no—
270C3×Dic5—22no—
271C3×Dic5—42no—
272C3×Dic5#11714yescomplemented
273C5×Dic3—22no—
274C5×Dic3—42no—
275C23×D3—51no—
276C23×D3C22×D511yescharacteristic
277C22×C3×C4—51no—
278C22×C3×C4—51no—
279C22×C3×C4—52no—
280C22×C3×C4C22×D512yes—
281C22×C3×C4C22×D511yescharacteristic
282C22×Dic3—51no—
283C22×Dic3—52no—
284C22×Dic3C22×D512yes—
285C22×Dic3C22×D511yescharacteristic
286C2×D12—54no—
287C2×D12—102no—
288C2×D12—102no—
289C2×C4×D3—22no—
290C2×C4×D3—22no—
291C2×C4×D3—54no—
292C2×C4×D3—102no—
293C2×C4×D3—102no—
294C2×C4×D3C4×D514yes—
295C3×C4⋊C4—102no—
296C3×C4⋊C4C22⋊D514yescomplemented
297C3×C42—54no—
298C3×C42—104no—
299(C2×D3)⋊C4—24no—
300(C2×D3)⋊C4—54no—
301(C2×D3)⋊C4—54no—
302(C2×D3)⋊C4—104no—
303(C2×D3)⋊C4—202no—
304(C2×D3)⋊C4—202no—
305C4⋊Dic3—22no—
306C4⋊Dic3—54no—
307C4⋊Dic3—104no—
308C4⋊Dic3—102no—
309C4⋊Dic3—202no—
310C4×Dic3—24no—
311C4×Dic3—104no—
312C23×C5—32no—
313C23×C5—61no—
314C23×C5—61no—
315C23×C5—61no—
316C23×C5—61no—
317C23×C5—61no—
318C23×C5—61no—
319C23×C5C23×D311yescharacteristic
320C22×D5—32no—
321C22×D5—61no—
322C22×D5—61no—
323C22×D5—61no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5—32no—
330C2×C4×C5—61no—
331C2×C4×C5—61no—
332C2×C4×C5—68no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5C2×C4×D312yes—
337C2×C4×C5C2×D1214yes—
338C2×C4×C5C2×D1212yes—
339C2×C4×C5#23512yes—
340C22⋊D5—68no—
341C2×Dic5—21no—
342C2×Dic5—21no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—32no—
347C2×Dic5—61no—
348C2×Dic5—61no—
349C2×Dic5—62no—
350C2×Dic5—62no—
351C2×Dic5—62no—
352C2×Dic5C2×C4×D312yes—
353C2×Dic5C2×C4×D312yes—
354C2×Dic5#23712yes—
355C2×Dic5#23912yes—
356C4×D5—68no—
357C22×D4—151no—
358C23×C4—31no—
359C23×C4—151no—
360C4×D4—158no—
361#117—64no—
362#117—304no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—151no—
366C2×C4⋊C4—152no—
367C2×C4⋊C4—151no—
368C2×C4⋊C4—152no—
369C2×C4⋊C4D3×D511yescharacteristic
370C2×C22⋊C4—32no—
371C2×C22⋊C4—151no—
372C2×C22⋊C4—151no—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C42—32no—
377C2×C42—51no—
378C2×C42—52no—
379C2×C42—152no—
380#95—52no—
381#95—152no—
382#95—152no—
383C2×C3×C5C2×#11711yescharacteristic
384C2×C3×C5C2×C4×D411yescharacteristic
385C2×C3×C5#52011yescharacteristic
386C2×C3×C5#54711yescharacteristic
387C2×C3×C5#55011yescharacteristic
388C2×C3×C5#55511yescharacteristic
389C2×C3×C5#55611yescharacteristic
390D15—42no—
391C5×D3—42no—
392C23×C3C23×D511yescharacteristic
393C22×D3—21no—
394C22×D3—21no—
395C22×D3—21no—
396C22×D3—21no—
397C22×D3—21no—
398C22×D3—21no—
399C22×D3—52no—
400C22×D3—101no—
401C22×D3—101no—
402C22×D3—101no—
403C22×D3—101no—
404C22×D3—101no—
405C22×D3—101no—
406C22×D3C2×C4×D512yes—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—52no—
410C2×C3×C4—52no—
411C2×C3×C4—52no—
412C2×C3×C4—52no—
413C2×C3×C4—101no—
414C2×C3×C4—101no—
415C2×C3×C4—102no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4#70912yes—
420C2×C3×C4C2×C22⋊D514yes—
421C2×C3×C4C2×C22⋊D512yes—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—52no—
428C2×Dic3—101no—
429C2×Dic3—101no—
430C2×Dic3—102no—
431C2×Dic3—102no—
432C2×Dic3—102no—
433C2×Dic3C2×C4×D512yes—
434D12—108no—
435C4×D3—28no—
436C4×D3—108no—
437C22×C5—62no—
438C22×C5—62no—
439C22×C5—62no—
440C22×C5—121no—
441C22×C5—121no—
442C22×C5—121no—
443C22×C5—121no—
444C22×C5C22×C4×D311yescharacteristic
445C22×C5C22×D1211yescharacteristic
446C22×C5C2×#23511yescharacteristic
447C22×C5C2×#23711yescharacteristic
448C22×C5C2×#23911yescharacteristic
449C22×C5#100911yescharacteristic
450C22×C5#101011yescharacteristic
451C2×D5—62no—
452C2×D5—62no—
453C2×D5—62no—
454C2×D5—121no—
455C2×D5—121no—
456C2×D5—121no—
457C2×D5—121no—
458C4×C5—24no—
459C4×C5—62no—
460C4×C5—122no—
461C4×C5C4×D1214yes—
462Dic5—22no—
463Dic5—42no—
464Dic5—62no—
465Dic5—122no—
466Dic5#89214yescomplemented
467C24—31no—
468C24—151no—
469C2×D4—154no—
470C2×D4—302no—
471C2×D4—302no—
472C22×C4—31no—
473C22×C4—34no—
474C22×C4—32no—
475C22×C4—51no—
476C22×C4—51no—
477C22×C4—52no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—151no—
481C22×C4—154no—
482C22×C4—152no—
483C22×C4—302no—
484C22×C4—302no—
485C22×C4C2×D3×D512yes—
486C22×C4C2×D3×D511yescharacteristic
487C4⋊C4—62no—
488C4⋊C4—102no—
489C4⋊C4—154no—
490C4⋊C4—304no—
491C4⋊C4—302no—
492C4⋊C4—602no—
493C4⋊C4#120314yescomplemented
494C22⋊C4—64no—
495C22⋊C4—154no—
496C22⋊C4—154no—
497C22⋊C4—304no—
498C22⋊C4—602no—
499C22⋊C4—602no—
500C42—54no—
501C42—64no—
502C42—104no—
503C42—304no—
504C3×C5#235411yescharacteristic, complemented, Hall
505C22×C3C22×C4×D511yescharacteristic
506C22×C3C2×#70911yescharacteristic
507C22×C3D5×#4111yescharacteristic
508C22×C3D5×#4111yescharacteristic
509C22×C3#414911yescharacteristic
510C22×C3#415011yescharacteristic
511C22×C3C22×C22⋊D511yescharacteristic
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—41no—
516C2×D3—41no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—102no—
520C2×D3—102no—
521C2×D3—102no—
522C2×D3—201no—
523C2×D3—201no—
524C2×D3—201no—
525C2×D3—201no—
526C3×C4—24no—
527C3×C4—54no—
528C3×C4—102no—
529C3×C4—202no—
530C3×C4C4×C22⋊D514yes—
531Dic3—22no—
532Dic3—42no—
533Dic3—102no—
534Dic3—202no—
535C2×C5—122no—
536C2×C5C2×C4×D1211yescharacteristic
537C2×C5C2×#89211yescharacteristic
538C2×C5#552311yescharacteristic
539C2×C5#555511yescharacteristic
540C2×C5#556311yescharacteristic
541C2×C5#557711yescharacteristic
542C2×C5#558411yescharacteristic
543D5—122no—
544C23—32no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—152no—
552C23—301no—
553C23—301no—
554C23—301no—
555C23—301no—
556C23—301no—
557C23—301no—
558C23C22×D3×D511yescharacteristic
559D4—308no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—32no—
563C2×C4—52no—
564C2×C4—52no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—68no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—101no—
574C2×C4—101no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—152no—
579C2×C4—301no—
580C2×C4—301no—
581C2×C4—308no—
582C2×C4—302no—
583C2×C4—302no—
584C2×C4—302no—
585C2×C4#677112yes—
586C2×C4C4×D3×D512yes—
587C2×C4C2×#120314yes—
588C2×C4C2×#120312yes—
589C2×C3D5×#11711yescharacteristic
590C2×C3#6585711yescharacteristic
591C2×C3#6586611yescharacteristic
592C2×C3#6586711yescharacteristic
593C2×C3C2×C4×C22⋊D511yescharacteristic
594C2×C3#6615811yescharacteristic
595C2×C3#6616811yescharacteristic
596D3—42no—
597D3—202no—
598C5#7363811yescharacteristic, complemented, Sylow
599C22—62no—
600C22—62no—
601C22—62no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22—302no—
607C22—302no—
608C22—302no—
609C22—601no—
610C22—601no—
611C22—601no—
612C22—601no—
613C22C2×#677111yescharacteristic
614C22C2×C4×D3×D511yescharacteristic
615C22#9202011yescharacteristic
616C22D5×#23711yescharacteristic
617C22D5×#23911yescharacteristic
618C22#9203511yescharacteristic
619C22C22×#120311yescharacteristic
620C4—24no—
621C4—54no—
622C4—62no—
623C4—102no—
624C4—122no—
625C4—202no—
626C4—302no—
627C4—602no—
628C4C4×#120314yes—
629C3#1061397211yescharacteristic, complemented, Sylow
630C2—122no—
631C2—602no—
632C2#1175367011yescharacteristic
633C2#1175373611yescharacteristic
634C2#1175374411yescharacteristic
635C2D5×#89211yescharacteristic
636C2C2×C4×#120311yescharacteristic
637C2#1175496511yescharacteristic
638C2#1175524511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1203C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2354
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#91781C2semidirect
2. C2×C5×#892C2semidirect
4. C3×C4⋊C4×Dic5C2semidirect
6. #11748834C2semidirect
7. C4×#6668C2semidirect
8. #11748799C2semidirect
9. #11748786C2semidirect
10. #11748730C2semidirect
11. #11748724C2semidirect
35. C4×#1203C4semidirect
46. C4⋊C4×Dic5D3semidirect
71. C3×C5×C4⋊C4D4semidirect
89. C2×#892D5semidirect
218. C5×C4⋊C4D12semidirect
272. C3×Dic5#117semidirect
296. C3×C4⋊C4C22⋊D5semidirect
466. Dic5#892semidirect
493. C4⋊C4#1203semidirect
504. C3×C5#2354semidirect
598. C5#73638semidirect
629. C3#10613972semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#917812
C2×C5×#8922
C2×C4×#12032
C3×C4⋊C4×Dic52
#117489082
#117488342
C4×#66682
#117487992
#117487862
#117487302
#117487242
#106139723
#736385